Question: The sum of two numbers is $109$, and their difference is $37$. What are the two numbers?
Solution: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 109}$ ${x-y = 37}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 146 $ $ x = \dfrac{146}{2} $ ${x = 73}$ Now that you know ${x = 73}$ , plug it back into $ {x+y = 109}$ to find $y$ ${(73)}{ + y = 109}$ ${y = 36}$ You can also plug ${x = 73}$ into $ {x-y = 37}$ and get the same answer for $y$ ${(73)}{ - y = 37}$ ${y = 36}$ Therefore, the larger number is $73$, and the smaller number is $36$.